Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and the center yellow has 4 vertices
TypeRegular 6-polytope
Schläfli symbol {4,34}
Coxeter-Dynkin diagrams

5-faces12 {4,3,3,3}
4-faces60 {4,3,3}
Cells160 {4,3}
Faces240 {4}
Vertex figure5-simplex
Petrie polygondodecagon
Coxeter groupB6, [34,4]

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.

Related polytopes

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes.

Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.

Cartesian coordinates

Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are


while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with -1 < xi < 1.


orthographic projections
Coxeter plane B6 B5 B4
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
Dihedral symmetry [6] [4]
3D Projections

6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.

Hexeract Quasicrystal structure orthographically projected
to 3D using the Golden Ratio.

Related polytopes

This polytope is one of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.


External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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